Exam 9: Conics, Parametric Curves, and Polar Curves

What do the parametric equations x = t² + 3t and y = t + 4 describe?

Find the arc length of x = u, y = , 0 $\le$ u $\le$ .

Find the area of the region bounded by the ellipse x = 7 cos $\theta$ , y = 9 sin $\theta$ .

The graph of the rose curve r = cos(n $\theta$ ) has n leaves if n is an even integer and 2n leaves if n is an odd integer.

Find equations of the three normal lines to the parabola given parametrically by the equations x(t) = , y(t) = 2t, which pass through the point P (3, 0).

Find the points of intersection of the polar curves r = 4(1 + cos $\theta$ ) and r(1 - cos $\theta$ ) = 3.

Find the equation to the ellipse for which (1, -1) is a focus, x - y = 3 is the corresponding directrix, and the eccentricity is 1/2.

Determine the points where the parametric curve x = - 3t, y = - 12t have horizontal and vertical tangents.

Find the equation of the tangent line to the curve at the given t. x = 2 cot t, y = 2 t at t =

Show that the total arc length of the lemniscate = cos(2 $\theta$ ) is 4 d $\theta$ .

A plane curve C is given parametrically by x = tan(t) - 2, y = sec(t), t (- , ).Find the Cartesian equation of the curve C.

Find the length of one arch of the cycloid x = a( $\theta$ - sin $\theta$ ), y = a(1 - cos $\theta$ ).

Find the area bounded by one petal of the flower r = a cos(5 $\theta$ ).

Find the slope of the curve x = 6t + 3, y = 2 - 7t when t = 5.

Find the length of the curve x = cos t + sin t, y = sin t - cos t, from t = to t = .

Find the equation of the tangent line to the curve at the given t. x = , y = at t = 1.

Find the area swept out by line segments from the origin to the curve r = tan $\theta$ from $\theta$ = 0 to $\theta$ = .

Describe the graph of the polar equation r = 6(sin $\theta$ + cos $\theta$ ).

Find the slope of the spiral curve r = $\theta$ at $\theta$ = .

Convert the polar equation r² = sec(2 $\theta$ ) to Cartesian coordinates and identify the curve it represents.